Question

The following table shows the frequency distribution of body weight (in gm) of fish in a pond.

Body weight (in gm)	15-15.9	16-16.9	17-17.9	18-18.9	19.19.9	20-20.9
No. of fish	2	4	8	6	6	4

Find the modal body weight of fish in the pond.

Answer 1

Here, the class intervals are inclusive type. For calculating the mode, we have to convert the class intervals to exclusive type by subtracting 0.05 from the lower limit and adding 0.5 to the upper limit of each interval. Then, we have to prepare the frequency distribution table as shown:

Original class	Class boundaries	Frequency
15 - 15.9	14.95 - 15.95	2
16-16.9	15.95 - 16.95	4$\to$f1
17-17.9	16.95 - 17.95	8$\to$fm
18-18.9	17.95 - 18.95	6$\to$f2
19.19.9	18.95 - 19.95	6
20-20.9	19.95 - 20.95	4

Here, the maximum frequency f_m is 8. So, the corresponding modal class is 16.95 - 17.95.
$\mathrm{Mode}=L+\left[\frac{f_m-f_1}{2f_m-f_1-f_2}\right]h$
Here, L = lower limit of the modal class = 16.95
f_m = maximum frequency = 8
f₁ = frequency of pre-modal class = 4
f₂ = frequency of post-modal class = 6
h = width of modal class = 1

$$\begin{gathered} \therefore \mathrm{Mode}=16.95+\left[\frac{8-4}{2\left(8\right)-4-6}\right]\times 1 \\ =16.95+\left[\frac{4}{16-10}\right] \\ =16.95+\frac{4}{6} \\ =16.95+0.66=17.61 \\ \therefore \mathrm{The}\mathrm{modal}\mathrm{body}\mathrm{weight}\mathrm{of}\mathrm{fish}\mathrm{in}\mathrm{the}\mathrm{pond}\mathrm{is}17.61\mathrm{gm}. \end{gathered}$$